If no external force acts on a system

Options

A. (a) velocity of centre of mass remains constant
B. (b) position of centre of mass remains constant
C. (c) acceleration of centre of mass remains non-zero and constant
D. (d) All of the above

Answer

To solve the question, we need to analyze the implications of having no external force acting on a system. Let's break it down step by step:

### Step 1: Understand the concept of external force
When we say that no external force acts on a system, it means that the net force acting on the system is zero. This is a crucial point because it directly relates to the motion of the center of mass.

**Hint:** Consider how forces affect motion. What happens when the net force is zero?

### Step 2: Apply Newton's second law
According to Newton's second law, the net force acting on an object is equal to the rate of change of momentum (F = dp/dt). If no external force is acting on the system, we can express this as:

\[ F_{net} = 0 \]

This implies that the change in momentum (dp) over time (dt) is also zero.

**Hint:** Think about what it means for momentum to remain constant. How does this relate to velocity?

### Step 3: Relate momentum to velocity
Momentum (p) is defined as the product of mass (m) and velocity (v):

\[ p = m \cdot v \]

If the momentum is constant and mass does not change, then the velocity must also remain constant. Thus, we can conclude:

\[ v = \text{constant} \]

**Hint:** If velocity is constant, what can you say about acceleration?

### Step 4: Analyze acceleration
Acceleration (a) is defined as the rate of change of velocity with respect to time:

\[ a = \frac{dv}{dt} \]

Since we have established that the velocity is constant, it follows that the change in velocity (dv) is zero. Therefore, the acceleration must also be zero:

\[ a = 0 \]

**Hint:** Consider the implications of constant velocity on the position of the center of mass.

### Step 5: Conclusion about the center of mass
If the velocity of the center of mass remains constant, it means that the position of the center of mass changes at a constant rate. However, the position itself does not remain constant; it continues to change linearly over time. Thus, we conclude:

1. The velocity of the center of mass remains constant.
2. The position of the center of mass does not remain constant (it changes).
3. The acceleration of the center of mass is zero.

Based on this analysis, the correct answer to the question is that the velocity of the center of mass remains constant.

### Final Answer:
The velocity of the center of mass remains constant.